Computing Position As A Function of Time
Kepler used his two first laws for computing the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.
The procedure for calculating the heliocentric polar coordinates (r,θ) to a planetary position as a function of the time t since perihelion, and the mean motion n = 2π/P, is the following four steps.
- 1. Compute the mean anomaly
- 2. Compute the eccentric anomaly E by solving Kepler's equation:
- 3. Compute the true anomaly θ by the equation:
- 4. Compute the heliocentric distance r from the first law:
The important special case of circular orbit, ε = 0, gives simply θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.
The proof of this procedure is shown below.
Read more about this topic: Kepler's Laws Of Planetary Motion
Famous quotes containing the words position, function and/or time:
“They who say that women do not desire the right of suffrage, that they prefer masculine domination to self-government, falsify every page of history, every fact in human experience. It has taken the whole power of the civil and canon law to hold woman in the subordinate position which it is said she willingly accepts.”
—Elizabeth Cady Stanton (1815–1902)
“The intension of a proposition comprises whatever the proposition entails: and it includes nothing else.... The connotation or intension of a function comprises all that attribution of this predicate to anything entails as also predicable to that thing.”
—Clarence Lewis (1883–1964)
“A sleeping man holds in a circle around him the thread of the hours, the order of years and of worlds. He consults them instinctively upon awaking and in one second reads in them the point of the earth that he occupies, the time past until his arousal; but their ranks can be mingled or broken.”
—Marcel Proust (1871–1922)